# Solving Rational Equations

A
**
rational equation
**
is an equation with
rational expressions
on either side of the equals sign.

**
ONE TECHNIQUE
**
for solving rational equations is
cross-multiplication
— what some textbooks call the
means/extremes property
.

This method works only if on each side of the equation there is only one rational expression.

**
Example 1:
**

Solve:

$\frac{7}{x\text{\hspace{0.17em}}+\text{\hspace{0.17em}}2}=\frac{x}{x\text{\hspace{0.17em}}+\text{\hspace{0.17em}}2}$

Cross multiplying, we get:

${x}^{2}+2x=7x+14$

This quadratic equation can be solved by factoring .

${x}^{2}-5x-14=0$

$(x-7)(x+2)=0$

**
Remember
**
to check in the original equation for validity of solutions. In this case,
$x=7$
is valid but
$x=-2$
isn't, since it means division by zero in the original equation.

**
ANOTHER METHOD
**
is to multiply through by the
least common denominator
of all of the fractions on either side of the equation.

**
Example 2:
**

Solve:

$\frac{x}{16}-\frac{3}{8x}=\frac{5}{16}$

The least common denominator (LCD) in this case is $16x$ . So, multiply both sides of the equation by $16x$ .

$\frac{x\left(16x\right)}{16}-\frac{3\left(16x\right)}{8x}=\frac{5\left(16x\right)}{16}$

${x}^{2}-6=5x$

Solve the quadratic equation by factoring.

${x}^{2}-5x-6=0$

$(x-6)(x+1)=0$

$x=6\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{or}\text{\hspace{0.17em}}\text{\hspace{0.17em}}x=-1$

**
Remember
**
to check back to make sure these solutions are valid – that is, that they don't result in division by zero when substituted in the original equation. In this case, both solutions are valid.